Question

计算行列式的值|a-100| |1b-10| |01c-1| |001d| 是四行四列的行列式

Answer 1

a -1 0 0
1 b -1 0
0 1 c -1
0 0 1 d r2+r1*1/a
=
a -1 0 0
0 b-1/a -1 0
0 1 c -1
0 0 1 d 按第1列展开
=
b-1/a -1 0
1 c -1
0 1 d *a r2-r3*1/d
=
b-1/a -1 0
1 c+1/d -1
0 0 d *a 按第3行展开
=ad*(b-1/a) *(c+1/d)
=(ab-1)*(cd+1)

追问

错了哦，第一步应该是r2+-1/ar1
答案是(1+ab)(1+cd)+ad

追答

不好意思……随手按错了字母
a -1 0 0
1 b -1 0
0 1 c -1
0 0 1 d r2-r1*1/a
=
a -1 0 0
0 b+1/a -1 0
0 1 c -1
0 0 1 d 按第1列展开
=
b+1/a -1 0
1 c -1
0 1 d *a c2-c3*1/d
=
b+1/a -1 0
1 c+1/d -1
0 0 d *a 按第3行展开
=ad* [(b+1/a) *(c+1/d) +1]
=(ab+1)*(cd+1) +ad
就是你要的答案